Homework 4 – STAT 543
Due Friday, February 10 by 5:00 pm (TA’s office);
you also may turn in the assignment in class on the same Friday
1. Problem 7.23, Casella and Berger (2nd Edition) (Problem 7.24 in 1st Edition.)
Note: assume that sample size n ≥ 3.
2. Problem 7.24, Casella and Berger (2nd Edition) (Problem 7.25 in 1st Edition.)
3. Read problem 7.62, Casella and Berger (2nd Edition). (Problem 10.8 in 1st Edition) Then show
(a) the posterior pdf of θ is normal with mean (1 − η)x̄n + ηµ and variance τ 2 η for η = σ 2 /(nτ 2 +
σ 2 ).
(b) Find the Bayes estimator of θ (under squared error loss)
(c) Then complete parts(a),(b),(c) of problem 7.62 in the text.
Note: part (b) from problem 7.62 follows from part(a) and the form of the Bayes estimator
4. Problem 10.1, Casella and Berger (2nd Edition)
Some extra problems: for practice, not to turn it
5. Read problem 7.63, Casella and Berger (2nd Edition). (Problem 10.9 in 1st Edition) You should be
able to solve this using the risk result from problem 7.62.
6. It turns out that, for the problem described in problem 7.62, (i.e., X1 , . . . , Xn iid N (θ, σ 2 ), σ 2
known), you can show that the sample mean X̄n is the minimax estimator of θ under the squared
2
error loss. To do so, we’ll use a result based on Problem 7.62: if T0µ,τ denotes the Bayes estimator
of θ based a N (µ, τ 2 ) prior for θ, then it holds that
2
“Bayes risk of T0µ,τ under N (µ, τ 2 ) prior” =
τ 2σ2
≤ min max RT (θ)
T
θ
nτ 2 + σ 2
(1)
where RT (θ) = Eθ (T − θ)2 is the risk of an estimator T = h(X1 , X2 , . . . , Xn ).1 To show that X̄ is
minimax, using the result in (1), try the following steps:
(a) Since (1) holds for any τ 2 > 0, show that
σ2
≤ min max RT (θ)
T
θ
n
must hold (ie, try taking limits).
(b) Show that the risk of the sample mean is constant: RX̄n (θ) =
σ2
n
for any θ.
(c) By (a)-(b), argue that X̄n must be the minimax estimator.
In defining minimax risk minT maxθ RT (θ), it is sometimes necessary to replace “min” and “max” with “inf” and “sup”
for technical reasons; this doesn’t change the nature of the problem though. If you are interested seeing how to show (1), please
talk with me; however, this is technical and not something you would be responsible for.
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